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Explicit estimates of some functions over primes

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Abstract

New results have been found about the Riemann hypothesis. In particular, we noticed an extension of zero-free region and a more accurate location of zeros in the critical strip. The Riemann hypothesis implies results about the distribution of prime numbers. We get better effective estimates of common number theoretical functions which are closely linked to \(\zeta \) zeros like \(\psi (x),\vartheta (x),\pi (x)\), or the \(k\mathrm{{th}}\) prime number \(p_k\).

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Acknowledgments

The author wishes to thank the anonymous reviewers for their helpful comments.

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  1. Pierre Dusart

    Present address: Département de Mathématiques, 123 avenue Albert Thomas, 87060, Limoges, France

Authors and Affiliations

  1. XLIM - UMR CNRS 7252, Faculté des Sciences et Techniques, Université de Limoges, 87060, Limoges, France

    Pierre Dusart

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  1. Pierre Dusart
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Correspondence to Pierre Dusart.

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Dusart, P. Explicit estimates of some functions over primes. Ramanujan J 45, 227–251 (2018). https://doi.org/10.1007/s11139-016-9839-4

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